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5. If R is a ring, then R[X], the set of all polynomials in X with coeﬃcients in R, is also a ring under ordinary polynomial addition and multiplication, as is R[X1 , . . , Xn ], the set of polynomials in n variables Xi , 1 ≤ i ≤ n, with coeﬃcients in R. Formally, the polynomial A(X) = a0 + a1 X + · · · + an X n is simply the sequence (a0 , . . , an ); the symbol X is a placeholder. The product of two polynomials A(X) and B(X) is a polynomial whose X k -coeﬃcient is a0 bk + a1 bk−1 + · · · + ak b0 .

We deﬁne multiplication on G componentwise: (h1 , k1 )(h2 , k2 ) = (h1 h2 , k1 k2 ). Since (h1 h2 , k1 k2 ) belongs to G, it follows that G is closed under multiplication. The multiplication operation is associative because the individual products on H and K are associative. The identity element in G is (1H , 1K ), and the inverse of (h, k) is (h−1 , k −1 ). Thus G is a group, called the external direct product of H and K. We may regard H and K as subgroups of G. More precisely, G contains isomorphic copies of H and K, namely H = {(h, 1K ) : h ∈ H} and K = {(1H , k) : k ∈ K}.

Belong to R and a1 ⊆ a2 ⊆ . . , then the sequence eventually stabilizes, that is, for some n we have an = an+1 = an+2 = . . (2) If R satisﬁes the ascending chain condition on principal ideals, then R satisﬁes UF1, that is, every nonzero element of R can be factored into irreducibles. (3) If R satisﬁes UF1 and in addition, every irreducible element of R is prime, then R is a UFD. Thus R is a UFD if and only if R satisﬁes the ascending chain condition on principal ideals and every irreducible element of R is prime.